Proving a quotient category is a localization in Mathlib via path objects

To show that a homotopy-quotient functor L : C ⥤ Ho is a localization at homotopy equivalences, use Functor.isLocalization_of_essSurj_of_full_of_exists_pathObjects (Mathlib.CategoryTheory.Localization.OfQuotient). You provide Full and EssSurj instances for L, show that W is inverted, and for each L.map f₀ = L.map f₁ give a PrepathObject Y with a RightHomotopy f₀ f₁ and W P.ι. There is a cylinder variant, but use the path-object version when the cylinder leaves the subcategory: for Kan complexes, X ⊗ Δ[1] is not Kan, while (ihom Δ[1]).obj Y is (Mathlib instance). For SSet, ι : Y ⟶ Y^Δ[1] is curry (snd Δ[1] Y), the evaluations are lift (const v_i) (𝟙 _) ≫ (ihom.ev Δ[1]).app Y, and a homotopy H gives curry ((β_ Δ[1] X).hom ≫ H.h). The homotopy ev₀ ≫ ι ~ 𝟙 needs min : Δ[1] ⊗ Δ[1] ⟶ Δ[1], defined as app _ := ↾fun x ↦ objMk (n := ⦋1⦌) ((objEquiv x.1).toOrderHom ⊓ (objEquiv x.2).toOrderHom) with naturality := rfl. Prove the identities on it as ext n x; apply objEquiv.injective; ext i : 3; exact min_eq_right (Fin.zero_le _). Check equalities of curried maps after apply uncurry_injective, then simp only [← Category.assoc, comp_lift] and a second simp only with the whiskerLeft_fst/snd and ι₀_fst/snd lemmas.